A circular sheet of metal (of negligible thickness) is cut into two sectors of angles \((1+t)\pi\) and \((1-t)\pi\) respectively, and each piece is bent into the form of a right circular cone by joining together its two bounding radii. If \(V(t)\) is the sum of the volumes of the two cones, prove that \(V(t)\) has a minimum when \(t=0\). Deduce, by general considerations, that \(V(t)\) is greatest when \(t=\pm t_0\), where \(t_0\) is a certain number satisfying \(0 < t_0 < 1\).
\(A\) is a fixed point on a sphere and \(P\) is a variable point on it. \(AP\) is produced to \(Q\) so that \(PQ\) is of constant length. Prove that the plane through \(Q\) perpendicular to \(PQ\) touches a fixed sphere.
Two light rods \(AB, BC\), each of length \(a\), are freely jointed at \(B\), and particles of masses \(m_1, m_2, m_3\) are attached at \(A, B, C\) respectively. The system is placed on a rough horizontal turntable, the particles alone making contact with it, so that \(A, B, C\) are at distances \(a, 2a, 3a\) respectively from the centre of rotation. Prove that, if the table rotates with constant angular velocity \(\omega\) and \[ a\omega^2(m_1+2m_2+3m_3) < \mu g(m_1+m_2+m_3), \] where \(\mu\) is the coefficient of friction at each contact, the system can remain upon the table without slipping.
If \(U = f(y/x)\) and \(U_n = r^n U\), where \(r^2 = x^2+y^2\), prove that \[ x\frac{\partial U}{\partial x} + y\frac{\partial U}{\partial y} = 0, \] \[ \frac{\partial^2 U_n}{\partial x^2} + \frac{\partial^2 U_n}{\partial y^2} = r^n\left(\frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} + \frac{n^2}{r^2}U\right). \] Hence, or otherwise, prove that, if the equation \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \] is satisfied by \(u = V(x,y)\), where \(V(x,y)\) is a homogeneous function of degree \(n\), then the equation is satisfied also by \(u = r^{-2n}V(x,y)\).
A side \(a\) and the opposite angle \(A\) of a triangle \(ABC\) are measured and found to be 6 inches and 30 degrees respectively, and the radius \(R\) of the circumcircle is calculated from these measurements. If each measurement is liable to a maximum error of 1 per cent. (in either direction), prove that the calculated value of \(R\) may be in error to the extent of about 1.9 per cent.
Prove that the polars of a fixed point \(A\) with respect to a system of confocal conics envelop a parabola touching the axes of the conics. Prove that the directrix of the parabola is the line joining \(A\) to the centre \(O\) of the confocals and that, if \(S\) is the focus of the parabola, the axes are the bisectors of the angle \(AOS\).
A smooth cylinder, whose normal cross section is a semi-circle of radius \(a\), is fixed with its plane face horizontal and in contact with the ground. A uniform chain lies in a small heap at the top of the cylinder, except for a length \(\frac{1}{4}\pi a\) which hangs down one side of the cylinder, the end just reaching the ground. The chain is released from rest. Assuming that each link is suddenly jerked into motion as the chain runs, show that, so long as a length \(x\) of the chain moves in contact with the cylinder, the velocity \(v\) of the chain satisfies the equation \[ \pi a v \frac{dv}{dx} + 2v^2 = 2ga, \] where \(x\) is the length of the chain heaped upon the ground. Hence show that \[ v^2 = ga(1-e^{-4x/\pi a}). \]
A uniform rod \(AB\) of mass \(m\) and length \(a\) can turn freely about a fixed point \(A\). A small ring of mass \(m'\) slides smoothly along the rod, and is attached by a light inelastic string of length \(b\) (\(b
A point \(Q\) is taken on the tangent at \(P\) to a plane curve \(\Gamma\) so that \(PQ\) is of fixed length. Prove that the normal at \(Q\) to the locus of \(Q\) when \(P\) moves along \(\Gamma\) passes through the centre of curvature of \(\Gamma\) at \(P\).
A point \(P\) of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] is joined to the points whose co-ordinates are \((\pm k, 0)\) and the joins meet the ellipse again in \(Q\) and \(R\). Prove that the pole of the line \(QR\) with respect to the ellipse lies on the ellipse \[ \frac{x^2}{a^4} + \frac{(a^2-k^2)^2}{(a^2+k^2)^2}\frac{y^2}{b^2} = 1. \]